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Worked fluid symbol: eigenvalues of the principal matrix for the radiation equation of state at an explicit timelike covector

Example

Take the smooth barotropic equation of state $p : J \to R$ , $J = (0, \infty)$ , $p (ϵ) = \frac{1}{3} ϵ$ , at any state $ϵ_{0} \in J$ . Then $ϵ + p = 4 ϵ / 3 > 0$ on $J$ and $p^{'} = 1 / 3 \in (0, 1)$ (so $c_{s} = 1 / \sqrt{3}$ ) and at a point choose a rest frame $n$ with spatial frame $(e_{1}, e_{2}, e_{3})$ and the future-directed timelike covector $ζ = - n^{♭} + \frac{1}{2} e_{1}^{♭}$ (in the notation $ζ_{0} = ζ (n) = 1$ , $ζ_{⊥} = \frac{1}{2} e_{1}$ , so $ζ_{0} = 1 > \frac{1}{2} = ∣ ζ_{⊥} ∣$ ). This example computes the matrix $A (ζ)$ of The barotropic fluid equations in enthalpy form are symmetric hyperbolic with positive principal symbol and verifies its eigenvalues and positive definiteness explicitly.

Facts & Assumptions

Given: the radiation equation of state on $J = (0, \infty)$ and a state $ϵ_{0} > 0$ , the rest frame and the covector $ζ = - n^{♭} + \frac{1}{2} e_{1}^{♭}$ , for which $ζ (n) = 1$ and $ζ_{⊥} = \frac{1}{2} e_{1}^{♭}$ .

[L1]

The principal symbol of the fluid system in the variables $(r, n^{1}, n^{2}, n^{3})$ at a covector $ζ$ is the symmetric matrix $A (ζ) = (\begin{array}{cc} ζ_{0} & c_{s} ζ_{⊥}^{T} \\ c_{s} ζ_{⊥} & ζ_{0} I_{3} \end{array})$ (The barotropic fluid equations in enthalpy form are symmetric hyperbolic with positive principal symbol clause (b)).

[L2]

$c_{s} = \sqrt{p^{'}} = \frac{1}{\sqrt{3}}$ for the radiation equation of state (The barotropic enthalpy variable r(epsilon): definition, strict monotonicity and the symmetric form of the fluid equations). [L2]

Verification

technique · direct

1.1L1L2algebra

(The matrix.) By [L1], [L2] and the chosen data, $A (ζ) = (\begin{array}{cccc} 1 & \frac{1}{2 \sqrt{3}} & 0 & 0 \\ \frac{1}{2 \sqrt{3}} & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array})$ .

2.1step 1.1algebra

(Eigenvalues.) The upper-left block $(\begin{array}{cc} 1 & \frac{1}{2 \sqrt{3}} \\ \frac{1}{2 \sqrt{3}} & 1 \end{array})$ has eigenvalues $1 \pm \frac{1}{2 \sqrt{3}}$ ; the remaining block is the identity with eigenvalue $1$ (multiplicity two). All three eigenvalues are positive: $1 - \frac{1}{2 \sqrt{3}} > 0$ because $2 \sqrt{3} > 1$ .

3.1step 2.1L1algebra∎

(Consistency with the general criterion.) The general criterion of [L1] gives positivity because $ζ_{0} = 1 > \frac{1}{2} = ∣ ζ_{⊥} ∣$ and $c_{s} < 1$ ; the explicit smallest eigenvalue $1 - \frac{1}{2 \sqrt{3}} \approx 0.711 > 0$ matches exactly the lower bound $ζ_{0} - c_{s} ∣ ζ_{⊥} ∣ = 1 - \frac{1}{2 \sqrt{3}}$ from that criterion.

Worked fluid symbol: eigenvalues of the principal matrix for the radiation equation of state at an explicit timelike covector

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

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